QUASILINEAR CONTROL: Performance Analysis and Design of Feedback Systems with Nonlinear Sensors and Actuators pot

QUASILINEAR CONTROL Performance Analysis and Design of Feedback Systems with Nonlinear Sensors and Actuators This is a textbook on quasilinear control (QLC). QLC is a set of methods for performance analysis and design of linear plant/nonlinear instrumentation (LPNI) systems. The approach of QLC is based on the method of stochastic linearization, which reduces the nonlinearities of actuators and sensors to quasilinear gains. Unlike the usual – Jacobian linearization – stochastic linearization is global. Using this approximation, QLC extends most of the linear control theory techniques to LPNI systems. In addition, QLC includes new problems, specific for the LPNI scenario. Examples include instrumented LQR/LQG, in which the controller is designed simultaneously with the actuator and sensor, and partial and complete performance recovery, in which the degradation of linear performance is either contained by selecting the right instrumentation or completely eliminated by the controller boosting. ShiNung Ching is a Postdoctoral Fellow at the Neurosciences Statistics Research Laboratory at MIT, since completing his Ph.D. in electrical engineering at the University of Michigan. His research involves a systems theoretic approach to anesthesia and neuroscience, looking to use mathematical techniques and engineering approaches – such as dynamical systems, modeling, signal processing, and control theory – to offer new insights into the mechanisms of the brain. Yongsoon Eun is a Senior Research Scientist at Xerox Innovation Group in Webster, New York. Since 2003, he has worked on a number of subsystem technologies in the xerographic marking processandimage registrationtechnology for theinkjet marking process. His interests are control systems with nonlinear sensors and actuators, cyclic systems, and the impact of multitasking individuals on organizational productivity. Cevat Gokcek was an Assistant Professor of Mechanical Engineering at Michigan State University. His research in the Controls and Mechatronics Laboratory focused on automo- tive, aerospace, and wireless applications, with current projects in plasma ignition systems and resonance-seeking control systems to improve combustion and fuel efficiency. Pierre T. Kabamba is a Professor of Aerospace Engineering at the University of Michigan. His research interests are in the area of linear and nonlinear dynamic systems, robust control, guidance and navigation, and intelligent control. His recent research activities are aimed at the development of a quasilinear control theory that is applicable to linear plants with nonlinear sensors or actuators. He has also done work in the design, scheduling, and operation of multi- spacecraft interferometric imaging systems, in analysis and optimization of random search algorithms, and in simultaneouspath planning andcommunication scheduling for UAVsunder the constraint of radar avoidance. He has more than 170 publications in refereed journals and conferences and numerous book chapters. Semyon M. Meerkov is a Professor of Electrical Engineering at the University of Michigan. He received his Ph.D. from the Institute of Control Sciences in Moscow, where he remained until 1977. He then moved to the Department of Electrical and Computer Engineering at the Illinois Institute of Technology and to Michigan in 1984. He has held visiting positions at UCLA (1978–1979); Stanford University (1991); Technion, Israel (1997–1998 and 2008); and Tsinghua, China (2008). He was the editor-in-chief of Mathematical Problems in Engineering, department editor for Manufacturing Systems of IIE Transactions, and associate editor of several other journals. His research interests are in systems and control with applications to production systems, communication networks, and the theory of rational behavior. He is a Life Fellow of IEEE. He is the author of numerous research publications and books, including Production Systems Engineering (with Jingshang Li, 2009). Quasilinear Control Performance Analysis and Design of Feedback Systems with Nonlinear Sensors and Actuators ShiNung Ching Massachusetts Institute of Technology Yongsoon Eun Xerox Research Center Webster Cevat Gokcek Michigan State University Pierre T. Kabamba University of Michigan Semyon M. Meerkov University of Michigan CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo, Delhi, Dubai, Tokyo, Mexico City Cambridge University Press 32 Avenue of the Americas, New York, NY 10013-2473, USA www.cambridge.org Information on this title: www.cambridge.org/9781107000568 © ShiNung Ching, Yongsoon Eun, Cevat Gokcek, Pierre T. Kabamba, and Semyon M. Meerkov 2011 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2011 Printed in the United States of America A catalog record for this publication is available from the British Library. Library of Congress Cataloging in Publication data Quasilinear control : performance analysis and design of feedback systems with nonlinear sensors and actuators / ShiNung Ching [et al.]. p. cm. Includes bibliographical references and index. ISBN 978-1-107-00056-8 (hardback) 1. Stochastic control theory. 2. Quasilinearization. I. Ching, ShiNung. QA402.37.Q37 2010 629.8  312–dc22 2010039407 ISBN 978-1-107-00056-8 Hardback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party Internet Web sites referred to in this publication and does not guarantee that any content on such Web sites is, or will remain, accurate or appropriate. To my parents, with love, SHINUNG CHING To my wife Haengju, my son David, and my mother Ahn Young, with love and gratitude, YONGSOON EUN To my family, with love and gratitude, PIERRE T. KABAMBA To my dear wife Terry and to our children, Meera, Meir, Leah, and Rachel, with deepest love and admiration, SEMYON M. MEERKOV Brief Contents Preface page xiii 1 Introduction 1 2 Stochastic Linearization of LPNI Systems 20 3 Analysis of Reference Tracking in LPNI Systems 66 4 Analysis of Disturbance Rejection in LPNI Systems 114 5 Design of Reference Tracking Controllers for LPNI Systems 134 6 Design of Disturbance Rejection Controllers for LPNI Systems 167 7 Performance Recovery in LPNI Systems 204 8 Proofs 225 Epilogue 275 Abbreviations and Notations 277 Index 281 vii Contents Preface page xiii 1 Introduction 1 1.1 Linear Plant/Nonlinear Instrumentation Systems and Quasilinear Control 1 1.2 QLC Problems 3 1.3 QLC Approach: Stochastic Linearization 4 1.4 Quasilinear versus Linear Control 5 1.5 Overview of Main QLC Results 9 1.6 Summary 14 1.7 Annotated Bibliography 14 2 Stochastic Linearization of LPNI Systems 20 2.1 Stochastic Linearization of Open Loop Systems 20 2.1.1 Stochastic Linearization of Isolated Nonlinearities 20 2.1.2 Stochastic Linearization of Direct Paths of LPNI Systems 29 2.2 Stochastic Linearization of Closed Loop LPNI Systems 30 2.2.1 Notations and Assumptions 30 2.2.2 Reference Tracking with Nonlinear Actuator 31 2.2.3 Disturbance Rejection with Nonlinear Actuator 36 2.2.4 Reference Tracking and Disturbance Rejection with Nonlinear Sensor 37 2.2.5 Closed Loop LPNI Systems with Nonlinear Actuators and Sensors 40 2.2.6 Multiple Solutions of Quasilinear Gain Equations 46 2.2.7 Stochastic Linearization of State Space Equations 50 2.3 Accuracy of Stochastic Linearization in Closed Loop LPNI Systems 53 2.3.1 Fokker-Planck Equation Approach 53 2.3.2 Filter Hypothesis Approach 55 ix x Contents 2.4 Summary 57 2.5 Problems 57 2.6 Annotated Bibliography 64 3 Analysis of Reference Tracking in LPNI Systems 66 3.1 Trackable Domains and System Types for LPNI Systems 67 3.1.1 Scenario 67 3.1.2 Trackable Domains and Steady State Errors 67 3.1.3 System Types 74 3.1.4 Application: Servomechanism Design 75 3.2 Quality Indicators for Random Reference Tracking in Linear Systems 79 3.2.1 Scenario 79 3.2.2 Random Reference Model 79 3.2.3 Random Sensitivity Function 81 3.2.4 Tracking Quality Indicators 86 3.2.5 Application: Linear Hard Disk Servo Design 88 3.3 Quality Indicators for Random Reference Tracking in LPNI Systems 90 3.3.1 Scenario 90 3.3.2 Saturating Random Sensitivity Function 92 3.3.3 Tracking Quality Indicators 98 3.3.4 Application: LPNI Hard Disk Servo Design 101 3.4 Summary 105 3.5 Problems 106 3.6 Annotated Bibliography 112 4 Analysis of Disturbance Rejection in LPNI Systems 114 4.1 Basic Relationships 114 4.1.1 SISO Systems 115 4.1.2 MIMO Systems 116 4.2 Fundamental Limitations on Disturbance Rejection 124 4.3 LPNI Systems with Rate-Saturated Actuators 125 4.3.1 Modeling Rate-Saturated Actuators 126 4.3.2 Bandwidth of Rate-Saturated Actuators 127 4.3.3 Disturbance Rejection in LPNI Systems with Rate-Saturated Actuators 128 4.4 Summary 130 4.5 Problems 132 4.6 Annotated Bibliography 133 5 Design of Reference Tracking Controllers for LPNI Systems 134 5.1 Admissible Pole Locations for Random Reference Tracking 134 5.1.1 Scenario 134 [...]... analysis In addition, the problem of stability of systems with saturating actuators has been addressed in numerous publications However, the issues of performance, that is, disturbance rejection and reference tracking, have been addressed to a much lesser extent These are precisely the issues considered in this volume and, therefore, we use the subtitle Performance Analysis and Design of Feedback Systems. .. problems, but from the point of view of design; both wide and narrow sense design problems are considered (problems P2 and P3) Chapter 7 addresses the issues of performance recovery (problems P4 and P5) Finally, Chapter 8 includes the proofs of all formal statements included in the book Each chapter begins with a short motivation and overview and concludes with a summary and annotated bibliography Chapters... describes the method of stochastic linearization as it applies to LPNI systems and derives equations for quasilinear gains in the problems of reference tracking and disturbance rejection Chapters 3 and 4 are devoted to analysis of quasilinear control systems from the point of view of reference tracking and disturbance rejection, respectively (problem P1) Chapters 5 and 6 also address tracking and disturbance... Systems with Nonlinear Actuators and Sensors In view of the above, one may ask a question: If all feedback systems include nonlinear instrumentation, how have controllers been designed in the past, leading to a plethora of successful applications in every branch of modern technology? The answer can be given as follows: In practice, most control systems are, indeed, designed ignoring the actuator and sensor... Intent and prerequisites: This volume is intended as a textbook for a graduate course on quasilinear control or as a supplementary textbook for standard graduate courses on linear and nonlinear control In addition, it can be used for self-study by practicing engineers involved in the analysis and design of control systems with nonlinear instrumentation The prerequisites include material on linear and nonlinear. .. nonlinearities f (·) and g(·), but also on all other elements of Figure 0.1, including the transfer functions and the exogenous signals, since, as it turns out, Na and Ns are functions of the standard deviations, σu and σy , of u and y, respectively, that is, ˆ ˆ ˆ ˆ Na = Na (σu ) and Ns = Ns (σy ) Therefore, we refer to the system of Figure 0.2 as a ˆ ˆ quasilinear control system Systems of this type are... Figure 1.2(c) takes place and then, similar to LC, develops rigorous methods for quasilinear closed loop systems analysis and design In both cases, of course, the analysis and design results are supposed to be used for the actual LPNI system of Figure 1.2(a) Which approach is better, LC or QLC? This may be viewed as a matter of belief or a matter of calculations As a matter of belief, we think that... gain (1.1) and illustrating it for typical nonlinearities of actuators and sensors, it concentrates on closed loop LPNI systems (Figure 1.2(a)) and their stochastic linearizations (Figure 1.2(c)) Since the quasilinear gain of an actuator, Na , depends on the standard deviation of the signal at its input, σu and, in turn, σu depends on Na , the quasilinear gain of the ˆ ˆ actuator is defined by a transcendental... equations derived ˆ in Chapter 2 are used throughout the book for various problems of performance analysis and design Chapter 3 is devoted to analysis of reference tracking in closed loop LPNI systems Here, the notion of system type is extended to feedback control with saturating actuators, and it is shown that the type of the system is defined by the plant poles at the origin (rather than the loop transfer... following: [1.45] R.C Booton, M.V Mathews, and W.W Seifert, Nonlinear servomechanisms with random inputs,” Dyn Ana Control Lab, MIT, Cambridge, MA, 1953 [1.46] R.C Booton, “The analysis of nonlinear systems with random inputs,” IRE Transactions on Circuit Theory, Vol 1, pp 32–34, 1954 [1.47] I.E Kazakov, “Approximate method for the statistical analysis of nonlinear systems, ” Trudy VVIA 394, 1954 (in Russian) . QUASILINEAR CONTROL Performance Analysis and Design of Feedback Systems with Nonlinear Sensors and Actuators This is a textbook on quasilinear. Engineering (with Jingshang Li, 2009). Quasilinear Control Performance Analysis and Design of Feedback Systems with Nonlinear Sensors and Actuators ShiNung